CLASS 22 CRAMER-RAO INEQUALITY скачать в хорошем качестве

CLASS 22 CRAMER-RAO INEQUALITY

The Cramér-Rao inequality is a fundamental concept in statistics that establishes a lower bound on the variance of any unbiased estimator for a parameter. It states that the variance of an unbiased estimator cannot be smaller than the reciprocal of the Fisher information, which is related to the likelihood function of the data. If an unbiased estimator's variance achieves this lower bound, it is called an efficient estimator or a minimum variance unbiased estimator (MVUE), providing the highest possible precision for a given parameter. Lower Bound on Variance: The inequality provides a theoretical minimum for the variance an unbiased estimator can have. Fisher Information: The lower bound is determined by the Fisher information, which quantifies the amount of information a sample contains about the parameter being estimated. Efficiency: The main purpose of the inequality is to assess the efficiency of an estimator. If an estimator reaches the Cramér-Rao lower bound, it is considered the most efficient possible unbiased estimator for that parameter. MVUE: When the Cramér-Rao inequality holds with equality, the estimator is a Minimum Variance Unbiased Estimator. Applications: It is a crucial tool for identifying and evaluating estimators in various fields, including signal processing, medicine, machine learning, and econometrics. In simpler terms: Imagine you're trying to estimate someone's height. The Cramér-Rao inequality tells you the best possible precision you can achieve, no matter how good your estimation method is, if your method doesn't systematically overestimate or underestimate the height (i.e., it's unbiased). If your method actually reaches that theoretical minimum precision, you've found the most accurate unbiased estimator possible.